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Let the domain be $[0,1]^2$. And let $W^x_t$ be the standard Brownian Motion started in $x\in [0,1]^2$ with absorbption on $\partial [0,1]^2$ and choose some $g\in \mathcal{C}^2((0,1)^2)\cap \mathcal{C}_0([0,1]^2)$. Now I would like to extend $g'$ and $g''$ to be continuous on the whole of $[0,1]^2$ and thus hope that I will be able to use Itô's Lemma. So basically I would like for the following to hold true:

$g(W_t)=\int_0^t \nabla g(W_s)d W_s +\int_0^t \Delta g(W_s)ds$

where $\nabla g$ and $\Delta g $ would need to be defined on the Boundary

There are two things troubling me:

  1. neither $\nabla g$ nor $\Delta g$ are defined on the Boundary (What would then be the best (most sensible) way of extending $g'$ and $g''$?)
  2. even if one was to define "Derivatives" at the boundary we would still be dealing with a somewhat different Space of Functions and I am not sure whether Itô's Lemma would still hold
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The question is unclear: what do you call to use Itô's lemma on a domain? – Did Jun 16 '12 at 9:59
Actually I meant to say to use Itô's Lemma on a function $g$ as described above (I will edit that - thank you for making me aware of it). To be more precise - let $X_t$ be a semimartingale and let $g\in \mathcal{C}^2((0,1)^2)\cap \mathcal{C}^0([0,1]^2)$ - could I then still apply Itô's Lemma to $g$ ? – Probabilitnator Jun 16 '12 at 13:26

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